Which rule helps identify possible rational zeros of a polynomial?

Which rule helps identify possible rational zeros of a polynomial?

• A. Rational Zero Theorem
• B. Factor Theorem
• C. Descartes' Rule of Signs
• D. Fundamental Theorem of Algebra

Answer: (A)

The main idea here is the Rational Zero Theorem. If a polynomial has integer coefficients, any rational zero must be of the form p/q where p is a divisor of the constant term and q is a divisor of the leading coefficient. This gives a finite list of candidate rational zeros to check, which is why this rule is used to identify possible rational zeros. Once you have that list, you can test each candidate by substitution or use the Factor Theorem to confirm which ones are actual zeros. Other rules don’t provide that explicit list: the Factor Theorem tells you when a specific number is a root but not which numbers to try, Descartes’ Rule of Signs limits how many positive or negative real roots there could be without naming them, and the Fundamental Theorem of Algebra guarantees the existence of roots but not which ones are rational.

The main idea here is the Rational Zero Theorem. If a polynomial has integer coefficients, any rational zero must be of the form p/q where p is a divisor of the constant term and q is a divisor of the leading coefficient. This gives a finite list of candidate rational zeros to check, which is why this rule is used to identify possible rational zeros. Once you have that list, you can test each candidate by substitution or use the Factor Theorem to confirm which ones are actual zeros. Other rules don’t provide that explicit list: the Factor Theorem tells you when a specific number is a root but not which numbers to try, Descartes’ Rule of Signs limits how many positive or negative real roots there could be without naming them, and the Fundamental Theorem of Algebra guarantees the existence of roots but not which ones are rational.